The goal of this project is to continue the principal investigator's research on analytic issues arising from four subareas: (i) the hydrodynamic flow of nematic liquid crystal materials, (ii) high dimensional phase-transition problem between two manifolds, (iii) conserved geometric motion of co-dimension two surfaces, and (iv) L-infinity variational problems. The first part of this project deals with the Ericksen-Leslie system modeling hydrodynamic flow of nematic liquid crystals, which is a strongly nonlinear-coupled system between the incompressible Navier-Stokes equation of the underlying fluid velocity field and the transported heat flow of harmonic maps for the orientation director field of the nematic liquid crystal molecules. The objective is to establish existence and partial regularity for Leray-Hopf type weak solutions in dimension three. The second project investigates the energy asymptotic of a singularly perturbed functional in the sense of Gamma-convergence and resolve the Keller-Rubinstein-Sternberg problem on the dynamics in terms of harmonic map heat flow under new boundary conditions and mean curvature flow of the sharp interface. The third project is to establish the local well-posedness of such a conserved mean curvature flow for generic initial surfaces. The fourth project is to study the uniqueness of general Aronsson's equations for Hamiltonian functions with spatial dependence and the regularity of infinity harmonic functions and Aronsson's equations corresponding to uniformly convex Hamiltonians.
The proposed problems in these areas are not only very challenging mathematically but also have strong connections and profound applications to other fields such as biology, chemical engineering, physics, fluid mechanics and material sciences. Mathematically, the nonlinear partial differential equations or systems involved in the project either are either highly degenerate elliptic problems or equations with super-critical nonlinearities whose resolutions will definitely contribute new ideas and techniques that will be useful in a variety of contexts. The hydrodynamic flow of nematic liquid crystals is among the most fundamental equations describing the dynamics of viscoelastic fluids and has its origination in LCD design and engineering. The conserved geometric motion has close connection with the Bose condensate physics. The L-infinity variational problems has found its applications in the optimal control, the image recovery engineering arise, the determination of optimal radiation treatments in chemotherapy, and the design of winning strategies for random game theories. Rigorous analysis of both the existence and the regularity of various solutions to such a system can predict the formation of singularities, allow researchers to gain insight into turbulent phenomena, and justify both computational and experimental studies made by applied scientists and engineers. This project will result in the publication of monographs and lecture notes from international summer schools for both researchers and graduate students, involve active training of advanced PhD students, and include the organization of specific conferences such as Ohio River Analysis Meetings, AMS and SIAM special sessions, and AIM or BIRS workshops.
